This new hydrodynamics is in fact of a much more general character. Many physically relevant wave phenomena exhibit complex spatiotemporal behaviours that cannot be restricted to macroscopically coherent dispersive shock waves.
In spite of integrability, the inherent randomness of many real-life systems (due to initial and boundary conditions or to complex mechanisms [1]}, [2]}) ultimately means results can only be obtained in statistical terms. This kind of random wave motion in nonlinear infinitely-dimensional Hamiltonian systems has beem dubbed `integrable turbulence' [3]} and recently attracted significant attention. At a microscopic scale associated with the system's coherence length, integrable dispersive hydrodynamics feature solitons—the localised nonlinear waves that exhibit particle-like properties such as elastic, pairwise interactions. The soliton-dominated integrable turbulence emerging at the macroscopic scale is often called soliton gas, a moniker that can be traced back to V. Zakharov's seminal paper in 1971 [4]}. It is in this work that he derived the kinetic equation for a “rarefied" gas of KdV solitons, by considering the modification of the soliton velocity due to the position shifts in
its pairwise collisions with other solitons in the gas. This kinetic picture was then generalized in 2003, taking the thermodynamic limit of spectral finite-gap solutions and their modulations, by G. El to KdV soliton gases of arbitrary density [5]}, and further in [6]} to soliton and breather gases of the focusing NLS equation. The spectral thermodynamic limit derivation in [5]}, [6]} has enabled in [9]}, [10]} a formal construction of kinetic equations for isotropic and anisotropic bidirectional soliton gases such as e.g., shallow-water soliton gas described by the Kaup-Boussinesq equation. Integrability of the spectral kinetic equation for soliton gas has been studied in [11]}, [12]} at the level of various hydrodynamic reductions. Beyond these important results in the mathematics of integrability, the physical relevance of soliton gases has been demonstrated in recent ocean observations [13]} and laboratory experiments [14]}, [15]}.
